IOTL, an example of gunboat diplomacy perpetrated by Britain, known in Brazilian sources as the "Christie Crisis", erupted against the Empire of Brazil in the early 1860's. The catalysts for the crisis revolved around the issue of extraterritorial justice:
-The marooning of the HMS Prince of Wales merchant ship at the southern Brazilian coast and its subsequent looting by a band of locals;
-And the arrest of three drunken British officers from the HMS Forte at Rio de Janeiro for civil disturbance (the officers were interred in a public prison of "lowly" status, provoking concern to British authorities).
In november of 1862, as a result of the ensuing crisis, the British admiral William D. Christie blockaded the harbour of Rio de Janeiro, demanding a resolution of the crisis on favorable terms to Britain, including the payment of an indemnity. Anti-British riots occured in the city, and emperor Pedro II refused to yield, believing in his country's sovereignty to be at stake. Brazil briefly cut diplomatic ties with Britain during the incident. Surprisingly, admiral Christie ended up yielding after finding out that Pedro did not, and the tide of the crisis ebbed until its resolution in 1865, at the beginning of the Paraguayan War.
It is speculated that the Brazilian invasion of Uruguay in support of the Colorado party was because of its feeling at the time of the crisis that its PR was at stake. The subsequent Brazilian-Colorado victory in the civil war is considered by many historians to have been the primary catalyst behind the decision of Paraguayan dictator Solano Lopez to declare war in 1865.
So, i ask: what if the "Christie Crisis" had instead resulted in war between Brazil and Great Britain?
HMS Prince of Wales merchant ship
HMS= Her/His Majesty's Ship, a prefix reserved for commissioned warships.

the British admiral William D. Christie blockaded the harbour of Rio de Janeiro
William Dougal Christie was the British minister plenipotentiary: Admiral Richard Laird Warren commanded the South-East Coast of America station. Christie did, however, write some exculpatory memoirs in 1865 which are available online.

The catalysts for the crisis revolved around the issue of extraterritorial justice
Slavery also underlay the crisis, and might well be relevant to its evolution:
'the British continued to press the point officially and unofficially... the larger aim in mind was the end of Negro slavery altogether. As early as 1856 the British minister was suggesting to the Brazilians that they must lessen their ties to this institution. It was Christie again who thrust at the vitals of the country to which he was accredited. In 1862 he wrote Lord Russell: "I have, on various occasions, suggested to your lordship the importance of endeavouring if possible to... persuade the Brazilian government to measures leading to the ultimate extinction of slavery, and in the meantime mitigating its evils".

His methods of persuasion were effective: in January 1863 Christie ordered reprisals against Brazilian shipping, and a break in diplomatic relations was the result. Minor and patently insignificant incidents were the pretext for this action; the real issues at stake were the uncounted Africans imported since 1831, the thousands of emancipados, and Brazilian slavery itself... Just one year after the reprisals were initiated and at a time when the British response to the ensuing rupture of diplomatic relations was still uncertain, the emperor urged the cabinet to being thinking about the future of slavery "so that the same thing will not happen to us as with respect to the slave trade". A similar fear of "force" was then expressed in the Senate.' ('Britain and the Onset of Modernization in Brazil 1850-1914,' Richard Graham)
So if Britain goes to war with Brazil partly over the issue of slavery, that seems to me that it will greatly reduce their chances of intervening on the Confederate side of the ACW, no? And knowing this, the behavior of both the Union and Confederacy might be different without the possibility of British aid.
I disagree with Graham there and I think he reads way too much into the Christie Crisis. Britain never cared about slavery, only the slave trade. If that was truly Britain’s intention they would not have issued an apology in 1865, but used the Paraguayan War to put even more pressure on Brazil. Had there been British pressure to end slavery, there’s no way Brazil could have held out until 1888 to abolish it altogether.

To answer the OP, I think a war would have been highly unlikely and it may go either way, depending on how much the British are willing to commit.